Consequences of Isoperimetry
A simple integration over r (or sum in the case of graphs) shows that a d-dimensional isoperimetric inequality implies a d-dimensional volume growth, namely
where B(x,r) denotes the ball of radius r around the point x in the Riemannian distance or in the graph distance. In general, the opposite is not true, i.e. even uniformly exponential volume growth does not imply any kind of isoperimetric inequality. A simple example can be had by taking the graph Z (i.e. all the integers with edges between n and n + 1) and connecting to the vertex n a complete binary tree of height |n|. Both properties (exponential growth and 0 isoperimetric dimension) are easy to verify.
An interesting exception is the case of groups. It turns out that a group with polynomial growth of order d has isoperimetric dimension d. This holds both for the case of Lie groups and for the Cayley graph of a finitely generated group.
A theorem of Varopoulos connects the isoperimetric dimension of a graph to the rate of escape of random walk on the graph. The result states
Varopoulos' theorem: If G is a graph satisfying a d-dimensional isoperimetric inequality then
where is the probability that a random walk on G starting from x will be in y after n steps, and C is some constant.
Read more about this topic: Isoperimetric Dimension
Famous quotes containing the words consequences of and/or consequences:
“War is thus divine in itself, since it is a law of the world. War is divine through its consequences of a supernatural nature which are as much general as particular.... War is divine in the mysterious glory that surrounds it and in the no less inexplicable attraction that draws us to it.... War is divine by the manner in which it breaks out.”
—Joseph De Maistre (1753–1821)
“Every expansion of government in business means that government in order to protect itself from the political consequences of its errors and wrongs is driven irresistibly without peace to greater and greater control of the nation’s press and platform. Free speech does not live many hours after free industry and free commerce die.”
—Herbert Hoover (1874–1964)